Topic 6 Hypothesis Testing

Question 1

One sample testing vs 2 sample testing

Answer 1

Single sample against population e.g grades in Birmingham vs population in England

Two samples against EACH OTHER. E.g grades in Birmingham vs London

Question 2

3 assumptions to conduct one sample hypothesis testing

Answer 2

Random sampling
Level of measurement is interval or ratio (to calc mean)
Sample distribution is normal (we can be sure of this if large size, using CLT)

Question 3

Step 1 of one sample hypothesis testing

Answer 3

State the null and alternate hypothesis

Question 4

Null hypothesis

Answer 4

Statement about the value of a population parameter

H0

Question 5

Alternate hypothesis h1

Answer 5

Statement accepted if null is false.

Inequalities are always part of the alternate i.e <,>

Question 6

Possible null and alternative hypothesis symbols

Answer 6

If h0 is = h1 is ≉
If h0 is greater or equal to, h1 is <
If h0 is less or equal to, h1 is >

Question 7

Steps to one sample hypothesis testing

Answer 7

1.State null and alternative

2.Choose level of significance e.g 95% confidence so alpha=0.05

3. Choose test statistic
4. Formulate decision rule (critical value is where the null hypothesis is rejected)
5. Make decisions
   - calculate test statistic

Find z value (sample mean (x bar) - population mean/(pop SD/root N (sample size)

E.g if critical region is 1.96 and z= 8, it falls in critical region so we reject null hypothesis

Question 8

What happens for one tailed tests

Answer 8

E.g left tail test, full 5% will lie in lower tale, changes the decision rule (critical value changes) now -1.65, 8>-1.65 so we now fail to reject null

Question 9

T statistic and z score formulas ( the value that will be compared to the critical region)

Answer 9

Z score

Sample mean (X bar) - population mean (μ)
/
σ/√N.

It is similar to Z value, but Z value doesn’t have root N, as no sample size.

T statistic

Same but s instead of σ

Question 10

When to use one vs two tailed

Answer 10

Want to know whether a different number of holidays happen in Edgbaston to Birmingham overall, use 2 tailed.

If we want to know whether edgbaston take MORE holidays then one tailed.

Question 11

When to use sigma or s

Answer 11

If sigma unknown use S!

Question 12

When to use z or t distribution, when unknown sigma

Answer 12

If large sample, Z
Small sample, T

Question 13

P-value

Answer 13

Highest level of confidence we can reject the null

Question 14

How to find p value example

Answer 14

Lower p value is, the more confident

If p value is 0.01, reject null up to 99% confidence.

Question 15

3 sample hypothesis test assumptions e.g grades London vs Birmingham

Answer 15

Two independent random samples used (not related)

Level of measurement is interval or ratio (to calc mean)

Normal sampling distribution (confirm if large, by CLT)

Question 16

Two sample hypothesis test steps (5)

Answer 16

State null and alternate: we state no difference between 2 samples for null.

Choose level of significance e.g alpha 0.05

Choose test statistic (z or t)

Formulate decision rule (find critical region)

Make decision

Question 17

Z score formula for two sample testing, if we know SD

Answer 17

(sample mean₁ - sample mean₂)-(μ₁ - μ₂)
/
√(σ²₁/n₁)+(σ²₂/n₂)

Question 18

Z score formula for two sample testing, if σ unknown.

Answer 18

Replace sigma1 and sigma2 (population deviation) with s1 and s2 (sample deviation)

Question 19

What do we use if sample is small in 2 sample test

Answer 19

T distribution

Question 20

If two populations have same SD, t statistic is….

Answer 20

Sample mean1(X bar1) - sample mean 2 (X bar 2)
/
√S²p (1/n₁ + 1/n₂)

S squared P is the pooled estimate of population variance.

Question 21

S²p formula

Answer 21

(n₁-1)s²₁ + (n₂-1)s²₂
/
(n₁+n₂-2)

n₁+n₂-2 is degrees of freedom e.g if n1=10 n2=9 DF=17

Question 22

If we assume 2 sample populations have different SDs unlike earlier , t statistic is…

2. Degrees of freedom parameter for this (different S.D)

Answer 22

Sample mean₁(X bar 1) - Sample mean₂ (X bar 2)
/
√s²₁/n₁ + s²₂/n₂

(Same denominator as large but sigma unknown!)

2. DF= (s²₁/n₁) + (s²₂/n2) all squared (AGAIN)
   /
   (S²₁/n₁)² / n₁-1 + (s²₂/n₂)²/n₂-1

Question 23

T statistic for dependent (paired) samples

Answer 23

d bar
/
S of d/√N

D bar is mean change in pair of observations.
S of d is SD of differences.
N is number of observations.

Question 24

S of D formula

Answer 24

√Σ(d-d bar)²
/
n-1

Question 25

One sampling and two Sampling test, when to use Z or T. Same rule for both

Answer 25

Z-when σ is known. Always use Z
If σ is unknown but sample is large, still use Z

T-when σ is unknown and sample is small.